I take the meaning here to be the grounds on which one can reasonably expect to have created some future benefit  aka good kamma.
Rhys Davids has "Three bases by merit accomplished, to wit, the bases composed of giving, of virtue, of study.";
Walshe has "grounds based on merit" meaning the same thing as Rhys Davids.

What's the difference?

In the first case the statement is that because one has done one or another of these types of deeds, one has a basis for expecting some future benefit;

In the case of Rhys Davids and Walshe, the statement is that one has created a basis for something unspecified by way of the merit of one or another of these deeds. I don't think that is the intent of the original nor, actually of Rhys Davids or Walshe. If we had a follow-up statement to the effect that by means of this bases states of further accomplishment might be gained, then this construction would make sense.

kiriya: PED: 1A. action, performance, deed; the doing=fulfilment. (MO: Often apparently serving the function of turning another word into one meaning the act of doing that: kusala- performance; dana- bestowing; papa- commission; mangala- celebration; sacchi- realization); B. an act in a special sense = promise, vow, dedication, intention, pledge; C. philosophically: action ineffective as to result, non-causative, an action which ends in itself; 2. making no difference, indefinite; of no result; indifferent, neither good nor bad and having no fruit of kamma

bhāvanā. Becoming, Development,  I think Walshe's "meditation," and Rhys Davids' "study" for bhavana are cases of what is being developed, but the word is more general and probably intended to be inclusive of all forms of development or self-development = becoming; or in the other sense of the word "living" meaning manner of living.